Solving Sudoku

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(No Transcript)
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What is Sudoku?

• A simple but addictive puzzle, invented in the
  USA in 1979 and called Number Place
• Became popular in Japan in 1986, where it was
  renamed Sudoku ( single number)
• First appeared in UK newspapers in 2004, and
  became an international craze in 2005.

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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 5 6 9 7 8 4 4
2 5
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 5 6 9 7 8 4 4
2 5
What number must go here?
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 5 6 9 7 8 4 4
2 5
1, as 2 and 3 already appear in this column.
1
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 1 5 6 9 7 8 4 4
2 5
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 1 5 6 9 7 8 4 4
2 5
3
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 1 5 3 6 9 7 8 4 4
2 5
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 1 5 3 6 9 7 8 4 4
2 5
2
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 1 3 8 5 7 6
1 3 9 8 1 2 5 7 3 1
8 9 8 2 1 5 2 3 6 9 7 8 4 4
2 5
And so on
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Example
Fill in the grid so that every row, column and
box contains each of the numbers 1 to 9
2 4 9 5 7 1 6 3 8 8 6 1 4 3 2 9 7 5 5 7 3 9 8 6 1
4 2 7 2 5 6 9 8 4 1 3 6 9 8 1 4 3 2 5 7 3 1 4 7 2
5 8 6 9 9 3 7 8 1 4 2 5 6 1 5 2 3 6 9 7 8 4 4 8 6
2 5 7 3 9 1
The unique solution for this easy puzzle.
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This Talk

• We show how to develop a program that can solve
  any Sudoku puzzle in an instant
• Start with a simple but impractical program,
  which is improved in a series of steps
• Emphasis on pictures rather than code, plus some
  lessons about algorithm design.

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Representing a Grid
type Grid Matrix Char type Matrix a
Row a type Row a a
A grid is essentially a list of lists, but
matrices and rows will be useful later on.
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Examples
empty Grid empty replicate 9 (replicate 9 '
')
easy Grid easy "2 1 38" ,
" 5" , " 7 6 " ,
" 13 " , " 981 257" ,
"31 8 " , "9 8 2 " ,
" 5 69784" , "4 25 "
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Extracting Rows
rows Matrix a ? Row a rows m m
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Columns
cols Matrix a ? Row a cols m transpose m
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And Boxes
boxs Matrix a ? Row a boxs m ltomittedgt
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Validity Checking
Let us say that a grid is valid if it has no
duplicate entries in any row, column or box
valid Grid ? Bool valid g all nodups (rows
g) ? all nodups (cols g) ?
all nodups (boxs g)
A direct implementation, without concern for
efficiency.
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Making Choices
Replace each blank square in a grid by all
possible numbers 1 to 9 for that square
choices Grid ? Matrix Char
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Collapsing Choices
Transform a matrix of lists into a list of
matrices by considering all combinations of
choices
collapse Matrix a ? Matrix a
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A Brute Force Solver
solve Grid ? Grid solve filter valid .
collapse . choices
Consider all possible choices for each blank
square, collapse the resulting matrix, then
filter out the valid grids.
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Does It Work?
The easy example has 51 blank squares, resulting
in 951 grids to consider, which is a huge number
4638397686588101979328150167890591454318967698009
Simple, but impractical!
gt solve easy ERROR out of memory
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Reducing The Search Space

• Many choices that are considered will conflict
  with entries provided in the initial grid
• For example, an initial entry of 1 precludes
  another 1 in the same row, column or box
• Pruning such invalid choices before collapsing
  will considerably reduce the search space.

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Pruning
Remove all choices that occur as single entries
in the corresponding row, column or box
prune Matrix Char ? Matrix Char
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And Again
Pruning may leave new single entries, so it makes
sense to iterate the pruning process
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And Again
Pruning may leave new single entries, so it makes
sense to iterate the pruning process
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And Again
Pruning may leave new single entries, so it makes
sense to iterate the pruning process
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And Again
Pruning may leave new single entries, so it makes
sense to iterate the pruning process
We have now reached a fixpoint of the pruning
function.
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An Improved Solver
solve Grid ? Grid solve filter valid
. collapse . fix prune . choices
For the easy example, the pruning process alone
is enough to completely solve the puzzle
Terminates instantly!
gt solve easy
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But
For a gentle example, pruning leaves around 381
grids to consider, which is still a huge number
443426488243037769948249630619149892803
No solution after two hours - we need to think
further!
gt solve' gentle
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Reducing The Search Space

• After pruning there may still be many choices
  that can never lead to a solution
• But such bad choices will be duplicated many
  times during the collapsing process
• Discarding these bad choices is the key to
  obtaining an efficient Sudoku solver.

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Blocked Matrices
Let us say that a matrix is blocked if some
square has no choices left, or if some row,
column, or box has a duplicated single choice
Key idea - a blocked matrix can never lead to a
solution.
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Expanding One Choice
Transform a matrix of lists into a list of
matrices by expanding the first square with
choices
expand Matrix a ? Matrix a
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Our Final Solver
solve Grid ? Grid solve search .
prune . choices
search Matrix Char ? Grid search m
blocked m complete m collapse m
otherwise g m' ? expand m
, g ? search (prune m')
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Notes

• Using fix prune rather than prune makes the
  program run slower in this case
• No need to filter out valid grids, because they
  are guaranteed to be valid by construction
• This program can now solve any newspaper Sudoku
  puzzle in an instant!

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The Result
This program has saved my life - my Sudoku
addiction is finally cured!!
Subliminal Message Haskell is the world's
greatest programming language.